patterns in proportional relationships block 1 student ...€¦ · student: class: date: patterns...

Student: Class: Date:

Patterns in proportional relationships Block 1 Student Activity Sheet

Copyright 2016 Agile Mind, Inc.® Content copyright 2016 Charles A. Dana Center, The University of Texas at Austin

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1. Blue and yellow can be used to make green. a. The mixtures of blue paint and yellow paint shown in this table create a particular

shade of green paint. List five other combinations of blue and yellow paint that will create the same shade of green paint.

Cups of blue paint Cups of yellow paint Ratio of blue to yellow

1 2 12

2 4 24

b. Plot your ratios on the coordinate plane. What patterns do you see in the graph?





2. REINFORCE Consider some of the proportional relationships you explored in the

Overview.

a. As the number of gallons used increases, how does the number of miles traveled change?

b. What is the relationship between the number of inches on the map and the number of actual miles?

c. In the original photograph, what is the relationship between the length and the width?





3. REINFORCE In a certain town, there are 2 houses for every 5 apartment units. Use this information to complete this activity.

a. Complete the table.

Number of houses Number of apartment units Ratio of houses to apartment units

2

10

15

8





b. Graph the data from your table.

c. Use your graph to predict the number of apartment units if there are 10 houses in the town.



Copyright 2016 Agile Mind, Inc. ® Content copyright 2016 Charles A. Dana Center, The University of Texas at Austin

Page 1 of 5 With space for student work

1. Consider a shade of green paint made by using 1 cup of blue paint for every 2 cups of yellow paint. a. Write a rule you can use to find the number of cups of yellow paint you will need for

any number of cups of blue paint. Write your rule in words.

b. The table represents the relationship between cups of blue paint and cups of yellow paint needed to make this shade of green. Complete the table. What process did you use to find the number of cups of yellow paint?

Cups of blue paint Process Cups of yellow paint

1

2

3

4

c. What is the coefficient in the algebraic rule?

d. Does this algebraic rule make sense for the scenario? Why or why not?

e. What would happen if you had 0 cups of blue paint? How many cups of yellow paint would you have?





f. How are the representations related?

2. Now consider a different mixture for making green paint. This paint mixture requires 3 cups of yellow paint for every cup of blue paint. a. Complete the table and graph for this paint mixture.

Cups of blue paint Process Cups of yellow paint

1

2

3

4





b. Write an algebraic rule that represents the relationship between the cups of blue paint, b, and the cups of yellow paint, y.

3. Use your work in question 2 to answer the following questions. REINFORCE

a. Using the algebraic rule you wrote, calculate how much yellow paint should be added for 8 cups of blue paint.

b. If you used 21 cups of yellow paint, how many cups of blue paint did you need?

4. Compare the graphs and algebraic rules for the two paint mixtures. Record similarities and differences.





5. In a proportional relationship, the constant of proportionality is the unit rate. Consider the two paint scenarios.

a. How can you identify the constant of proportionality in a graph?

b. How can you identify the constant of proportionality in an algebraic rule?

c. How can you identify the constant of proportionality in a table?





6. These representations show the relationship between blue paint and yellow REINFORCEpaint for different shades of green. For each representation, determine the constant of proportionality and write an algebraic rule.

a. b.

c. Cups

of blue paint

Cups of yellow paint

Ratio

3 2

6 4

9 6

12 8





1. Each medium cheese pizza costs $7.00. a. Complete the table and graph for this pizza relationship.

Number of pizzas Process Cost in

dollars

0

1

2

3

4

b. Find an algebraic rule that represents the relationship between the number of pizzas, n, and the total cost, C.

2. REINFORCE Use your work in question 1 to answer the following questions.

a. Using the algebraic rule you wrote, calculate how much it would cost to buy 10 pizzas.

b. If you have exactly $84 to spend, how many pizzas can you order?

c. What if you buy 0 pizzas? How much would that cost? Does your answer make sense?





3. Cheese pizzas cost $7.00 each, and there is a one-time charge of $5.00 to have your pizzas delivered. Answer the following questions about this situation. a. Complete the table and graph for this pizza and delivery charge relationship.

Number of pizzas Process Cost in

dollars

1

2

3

4

b. Find an algebraic rule that represents the relationship between the number of pizzas,

n, and the total cost, C.

4. REINFORCE Use your work in question 3 to answer the following questions.

a. How much would it cost to buy 10 pizzas and have the pizzas delivered?

b. If you have exactly $75 to spend in this situation, how many pizzas can you order?

c. What if you buy 0 pizzas? How much would that cost? Does your answer make sense? Why or why not?





5. Compare the two pizza problems. a. How are the pizza situations similar and how are they different?

Similar Different

b. How are the tables of data for the pizza problems similar and how are they different?

Similar Different

c. How are the algebraic rules similar and different?

Similar Different

d. How are the graphs of the data similar and how are they different?

Similar Different

e. What values make sense as inputs for the algebraic rules if you allow only numbers

that make sense in the context of the situation? Pizzas without delivery Pizzas with delivery charge





6. Consider the following question: For which situation can you just multiply REINFORCEthe input value by a constant to get an output value? To investigate this question, test a few input-output pairs. Start by thinking about the pizza situation without a delivery charge.

a. When you double the number of pizzas you order, what happens to the cost of the

order? Give specific examples.

b. When you triple the number of pizzas you order, what happens to the cost of the

order? Give specific examples.

c. How does the number of pizzas you can order for $42 compare to the number of

pizzas you can order for $14?

d. For this pizza situation, can you just multiply the input value by a constant to get the output value?





7. Answer these questions about pizzas with delivery. REINFORCE

a. When you double the number of pizzas you order, does the cost of the order double?

Give specific examples.

b. When the cost of the order is cut in half, is the number of pizzas you order also cut in half? Give specific examples.

c. For this pizza situation, can you just multiply the input value by a constant to get the

output value?





1. Consider the tables representing the cost of pizzas with delivery and the cost of pizzas without a delivery charge.

Pizzas without delivery Pizzas with delivery Number of pizzas

Cost in dollars Ratio Number

of pizzas Cost in dollars Ratio

0 0 0 5

1 7 1 12

2 14 2 19

3 21 3 26

4 28 4 33

a. Complete the table for pizzas with delivery by filling in the ratios. What do you notice

about these ratios?

b. Complete the table for pizzas without delivery by filling in the ratios. What do you

notice about these ratios?





2. Compare the graphs and algebraic rules of the two pizza scenarios.

a. How can you identify the rate of change using these graphs and algebraic rules?

b. For the proportional situation, how can you identify the constant of proportionality?





3. REINFORCE The values in the tables represent points on lines.

A B C D x y x y x y x y 0 4 0 0 0 7 0 0 1 11 1 7 1 7.50 1 2 2 18 2 14 2 8.00 2 4

a. When a relationship is proportional, you can multiply by the input value to get the

output value. Which tables represent proportional relationships?

b. When a relationship is non-proportional, you cannot just multiply by the input value to get the output value. Which tables represent relationships that are non-proportional?

c. Graph the points from the tables. Draw a different colored line through each set of

points. Label the lines A, B, C, and D, corresponding to the tables.





d. Which graphs represent relationships that are proportional? How can you tell by looking at the graph?

4. REINFORCE Mario’s Pizza sells medium pizzas for $5.50 and $1.50 per topping.

a. Fill in the table. Then graph the points.

Number of toppings, n

Cost in dollars, C

0

1

b. Write an algebraic rule relating the price of the pizza to the number of toppings.

c. Does this situation represent a proportional or non-proportional relationship? Explain.





5. Mario’s Pizzas sells large cheese pizzas for $9.00 and will deliver locally for a REINFORCEcharge of $3.00. a. Fill in the table. Then graph the points.

Number of pizzas, n

Cost in dollars, C

1

b. Write an algebraic rule relating the cost to the number of pizzas.

c. Does this situation represent a proportional or non-proportional relationship? Explain.





1. A landscaper needs to determine how many tiles will surround a 4-foot x 4-foot pond. How can you figure it out, without counting each individual tile? List some possible ways.

2. Draw several solutions you found using the animation. Write the corresponding number sentence in the table.

Solution strategy Number sentence





3. When you have finished drawing three more square ponds, answer the following questions. a. Create a table and a graph to represent the relationship between side length of a

pond and number of tiles needed to surround the pond. Use the data from the four pond dimensions you explored.





b. Write an algebraic rule that represents the relationship between the side length of a pond and the number of tiles needed to surround the pond. What do each of the numbers and letters in your rule represent?

c. Write one of the algebraic rules that your class discussed. What do the parts of the rule represent?

d. Is the relationship in the graph proportional or non-proportional? How do you know? What is the constant of proportionality?

e. What does the point (0,4) represent? What does the point (1,8) represent? Do these points make sense, given the scenario?





1. Create a table, a graph, and an algebraic rule to represent the relationship of pond side length, s, to amount of fencing, f.

2. Compare the tables, graphs, and algebraic rules for the pond tiling and pond fencing

scenarios. Is either relationship proportional?





3. If you found a proportional relationship in question 5, identify the constant of proportionality using each of the representations. Explain how you found it in each case.

a. In the table:

Side length of

pond (s) Amount of fencing (f) Ratio

1 4 2 8 3 12 4 16 s f = 4s

b. In the graph:

c. In the algebraic rule:





4. Complete the statements to show what you have learned about proportional relationships.

constant of proportionality line intercept

origin rate of change ratios

5. Consider this image created with the 4 circles. REINFORCE

a. Write an expression that describes the total number of intersections in the image.





b. Now write an expression that describes the total number of intersections there would be if the pattern continued for 7 circles. Draw out the pattern to see if your expression is correct.

c. Write a general expression for the total number intersections in a pattern with n circles.

6. Consider the pattern in the first three stages of this tile model. REINFORCE

Stage 1 Stage 2 Stage 3

a. Write an expression for the number of tiles that will be in stage 4.

b. Is the number of tiles in a stage proportional to the stage number? Explain why or why not.





1. Carlos is purchasing a cell phone plan. The company offers three options for REINFORCEits monthly service.

a. Carlos thinks he will use about 200 minutes a month. Calculate how much each of these options would cost Carlos for 1 month.

b. Which of these options is the least expensive for Carlos?

c. How many minutes each month would make the cost of the Regular Plan and the

All in One the same? Explain how you found your answer.

Pay as You Go

Pay 60¢ each minute.

Regular Plan

Pay $40 a month and 20¢ each minute.

All in One

Pay just $100 per month for

unlimited minutes.





d. Construct tables and graphs of all three options.





e. Describe how the graphs are different, and why this occurs.

f. Identify which phone plan relationship, if any, is

i. proportional:

ii. non-proportional:

g. If any of the relationships are proportional, identify the constant of proportionality.





2. Sonya’s favorite pizza parlor is called Mega-Toppings Pizza. Its name is fitting REINFORCEsince it advertises 35 different toppings.

a. Sonya thinks she would like the large cheese pizza with the following toppings: pepperoni, salami, olives, onions, green peppers, and shrimp. Determine the total cost of her pizza.

b. What is the cost of the most expensive large pizza you can buy at Mega-Toppings?

Show how you figured it out.

c. If t stands for number of toppings on a large cheese pizza and C stands for total cost, find an algebraic rule that gives C in terms of t.

d. Is the relationship between the cost and the number of toppings proportional or non-proportional? How do you know?

Mega-Toppings Pizza

Price Menu Large Cheese Pizza $8.00 48¢ for each topping*

*35 different toppings available





e. Toni ordered a large cheese pizza from Mega-Toppings. The pizza cost $13.28. How many toppings did Toni order? Show your calculations.

f. Which of these graphs shows the relationship between the number of toppings, t, and the cost of the pizza, C? Explain how you figured it out.

t

C

t

C

t

C

t

C Graph 1

Graph 4 Graph 3

Graph 2





3. Jake makes staircases with toothpicks arranged in squares. He counts the REINFORCEnumber of toothpicks he uses and makes a table to show his findings.

a. How many toothpicks are on the perimeter of Staircase 17? Explain how you figured it out.

b. Write an algebraic rule to represent the relationship between the staircase number

and the perimeter of the staircase.





c. Is the relationship between the staircase number and the number of toothpicks proportional or non-proportional? How do you know? If so, what is the constant of proportionality?

4. REINFORCE Cherie makes a tile pattern in many different sizes. In her pattern, white

tiles surround gray tiles. Here are her first three patterns.

a. In the space next to Pattern 3, draw a diagram to show Pattern 4.

b. Cherie makes a table to show how many tiles are needed to make different sizes of her pattern. Complete the table.

Pattern number Number of white tiles

1 8 2 10 3 4 5 6





c. Explain how you determined the number of white tiles needed for Pattern 6.

d. On the grid, plot the results from the pattern.

e. How many white tiles are needed for Pattern 24? Explain your answer.





f. What is the largest pattern Cherie can make using 30 white tiles? Explain your answer.

g. Is the relationship between the number of tiles and the pattern number proportional or non-proportional? How do you know?


Patterns in proportional reasoning Literacy Task



Template Task Collection 2 Copyright Literacy Design Collaborative, June 2013

After reading and investigating the topic Patterns and proportional relationships, write a brief essay in which you compare proportional linear relationships and non-proportional linear relationships and argue how you can tell whether a given relationship is proportional. Support your position with evidence from the texts.

General outline: Draft 1: Peer Review by _________________ (name) on ____________(date). Draft 2 (optional): Peer Review by _________________ (name) on ____________(date)


Patterns in proportional relationships Literacy Task Rubric


Page 1 of 1 Template Task Collection 2 Copyright Literacy Design Collaborative, June 2013

Argumentation Teaching Task Rubric for Template Task Collection 2

Scoring Elements

Not Yet Approaches Expectations Meets Expectations Advanced 1 1.5 2 2.5 3 3.5 4

Focus Attempts to address prompt, but

lacks focus or is off-task.

Addresses prompt appropriately and establishes a position, but focus is uneven. D. Addresses

additional demands superficially.

Addresses prompt appropriately and maintains a clear, steady focus. Provides a generally convincing

position. D: Addresses additional demands sufficiently

Addresses all aspects of prompt appropriately with a consistently strong

focus and convincing position. D: Addresses additional demands with

thoroughness and makes a connection to claim.

Controlling Idea

Attempts to establish a claim, but lacks a clear purpose. Establishes a claim. Establishes a credible claim.

Establishes and maintains a substantive and credible claim or proposal.

Reading/ Research

Attempts to reference reading materials to develop response,

but lacks connections or relevance to the purpose of the

prompt.

Presents information from reading materials relevant to

the purpose of the prompt with minor lapses in accuracy or

completeness.

Accurately presents details from reading materials relevant to the purpose of the prompt to develop

argument or claim.

Accurately and effectively presents

important details from reading materials to develop argument or claim.

Development

Attempts to provide details in response to the prompt, but

lacks sufficient development or relevance to the purpose of the

prompt.

Presents appropriate details to support and develop the focus, controlling idea, or claim, with minor lapses in the reasoning,

examples, or explanations.

Presents appropriate and sufficient details to support and develop the focus, controlling idea, or claim.

Presents thorough and detailed information to effectively support and develop the focus, controlling idea, or

claim.

Organization Attempts to organize ideas, but

lacks control of structure.

Uses an appropriate organizational structure for

development of reasoning and logic, with minor lapses in

structure and/or coherence.

Maintains an appropriate organizational structure to address

specific requirements of the prompt. Structure reveals the reasoning and

logic of the argument.

Maintains an organizational structure that intentionally and effectively enhances

the presentation of information as required by the specific prompt.

Structure enhances development of the reasoning and logic of the argument.

Conventions

Attempts to demonstrate standard English conventions,

but lacks cohesion and control of grammar, usage, and mechanics.

Sources are used without citation.

Demonstrates an uneven command of standard English

conventions and cohesion. Uses language and tone with

some inaccurate, inappropriate, or uneven

features. Inconsistently cites sources.

Demonstrates a command of standard English conventions and cohesion, with few errors. Response includes

language and tone appropriate to the audience, purpose, and specific

requirements of the prompt. Cites sources using appropriate format with

only minor errors.

Demonstrates and maintains a well-developed command of standard English

conventions and cohesion, with few errors. Response includes language and

tone consistently appropriate to the audience, purpose, and specific

requirements of the prompt. Consistently cites sources using appropriate format.

Content Understanding

Attempts to include disciplinary content in argument, but

understanding of content is weak; content is irrelevant, inappropriate, or inaccurate.

Briefly notes disciplinary content relevant to the

prompt; shows basic or uneven understanding of content;

minor errors in explanation.

Accurately presents disciplinary content relevant to the prompt with

sufficient explanations that demonstrate understanding.

Integrates relevant and accurate disciplinary content with thorough

explanations that demonstrate in-depth understanding.

patterns in proportional relationships block 1 student ...€¦ · student: class: date: patterns...

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